delamination of a strong film from a ductile substrate during indentation unloading

Delamination of a strong film from a ductile substrate during
indentation unloading

A. Abdul-Baqi and E. Van der Giessen
Delft University of Technology, Koiter Institute Delft, Mekelweg 2, 2628 CD Delft, The Netherlands

(Received 23 October 2000; accepted 23 February 2001)

In this work, a finite element method was performed to simulate the spherical
indentation of a ductile substrate coated by a strong thin film. Our objective was to
study indentation-induced delamination of the film from the substrate. The film was
assumed to be linear elastic, the substrate was elastic–perfectly plastic, and the indenter
was rigid. The interface was modeled by means of a cohesive surface. The constitutive
law of the cohesive surface included a coupled description of normal and tagential
failure. Cracking of the coating itself was not included. During loading, it was found
that delamination occurs in a tangential mode rather than a normal one and was
initiated at two to three times the contact radius. Normal delamination occurred during
the unloading stage, where a circular part of the coating, directly under the contact
area was lifted off from the substrate. Normal delamination was imprinted on the load
versus displacement curve as a hump. There was critical value of the interfacial
strength above which delamination was prevented for a given material system and a
given indentation depth. The energy consumption by the delamination process was
calculated and separated from the part dissipated by the substrate. The effect of
residual stress in the film and waviness of the interface on delamination was discussed.

I. INTRODUCTION displacement curves show a reduction in the stiffness or
Industrial application of thin hard-film-coated systems even a sudden discontinuity which is usually attributed to
continuously progresses. Coatings are commonly used to the coating cracking. Delamination without any accom-
enhance reliability, such as chemical resistance, wear re- panying through-thickness cracks has been observed by
sistance, corrosion resistance, and thermal barriers. Ad- Li and Bhushan2 in their nanoindentation experiments on
hesion between the film and the substrate determines, to single and multilayer coatings. There is no evidence in
a great deal, the durability of that system. The enhance- the literature, to the authors’ knowledge, whether delami-
ment gained by the coating may be accompanied by the nation can give rise to any characteristic fingerprint on
risk of poor adhesion between the coating and the sub- the load versus displacement curve.
strate. Failure of the interface between the coating and Bagchi and Evans3 have reviewed the mechanics of
the substrate may lead to premature failure of otherwise thin film decohesion motivated by residual stress. The
long lasting systems. Indentation is one of the traditional emphasis in their work is on the role of the interface
methods to quantify the mechanical properties of mate- debond energy and the methods of its quantitative meas-
rials, and during the last decades it has also been advo- urement. They argue that most thin film adhesion tests do
cated as a tool to characterize the properties of thin films not measure the interface debond energy because the
or coatings. At the same time, for example for hard wear- strain energy release rate cannot be deconvoluted from
resistant coatings, indentation can be viewed as an el- the work done by the external load. Viable procedures to
ementary step of concentrated loading. For these reasons, extract the interfacial energy from indentation experi-
many experimental as well as theoretical studies have ments will depend strongly on the precise mechanisms
been devoted to indentation of coated systems during involved. The relative contribution of each mechanism to
recent years. the overall observed behavior and failure mode depends
Interfacial delamination is commonly observed in on the material properties and loading conditions in a
indentation experiments to be accompanied by other complex manner. In the case of ductile films on a
failure phenomena, such as coating cracking and subse- hard substrate, coating delamination is coupled to plas-
quent spalling. 1,2 The corresponding load versus tic expansion of the film with the driving force for

1396 J. Mater. Res., Vol. 16, No. 5, May 2001 © 2001 Materials Research Society

A. Abdul-Baqi et al.: Delamination of a strong film from a ductile substrate during indentation unloading

delamination being delivered via buckling of the film II. PROBLEM FORMULATION
(see also Ref. 4). On the other hand, coatings on rela- The interface between the coating and the substrate is
tively ductile substrates often fail during indentation by modeled by means of a cohesive surface, where a small
radial and in some cases circumferential cracks through displacement jump between the film and substrate is al-
the film. The mechanics of delamination in such systems lowed, with normal and tangential components n and
has been analyzed by Drory and Hutchinson5 for deep
t, respectively. The interfacial behavior is specified in
indentation with depths that are 2 to 3 orders of magni- terms of a constitutive equation for the corresponding
tude larger than the coating thickness. They have also traction components Tn and Tt at the same location.
reviewed briefly the commonly used test methods for The constitutive law we adopt in this study is an elastic
evaluating adhesion. one, so that any energy dissipation associated with sepa-
Hainsworth et al.6 have suggested a simple model for ration is ignored. Thus, it can be specified through a
estimating the work of interfacial debonding from the potential, i.e.,
maximum indentation depth and the final delamination
radius. In this model, the elastic energy of the indented
coating is approximated by the elastic energy of a cen- T =− = n, t . (1)
trally loaded disc. The idea has also been used in cross-
sectional indentation by Sa nchez et al. 7 as a new
´
technique to characterize interfacial adhesion. The pro- The potential reflects the physics of the adhesion be-
portionality between the delamination area and the film tween coating and substrate. Here, we use the potential
lateral deflection predicted by the model was confirmed that was given by Xu and Needleman9
by the experimental results.
The objective of the present paper is to offer an im- n n 1−q
proved understanding of indentation-induced delamina- = + exp − 1−r+
n n
n n r−1
tion and to test the validity of the above-mentioned
simple estimates. For this purpose, we perform a numeri- r−q n t
2
− q+ exp − . (2)
cal simulation of the process of indentation of thin elastic r−1 n t
2

film on a relatively soft substrate with a small spherical
indenter. The complete cycle of the indentation process, with n and t the normal and tangential works of
both loading and unloading, is simulated. The indenter is separation (q t/ n) and n and t two characteris-
assumed to be rigid, the film is elastic and strong, and the tics lengths. The parameter r governs the coupling
substrate is elastic–perfectly plastic. The interface is between normal and tangential responses. As shown in
modeled by a cohesive surface, which allows one to Fig. 2, both tractions are highly nonlinear functions
study initiation and propagation of delamination during of separation with a distinct maximum of the nor-
the indentation process. Separate criteria for delamina- mal (tangential) traction of max ( max) which occurs at
tion growth are not needed in this way. The aim of this a separation of n n ( t t/√2). The normal
study is to investigate the possibility and the phenom-
enology of interfacial delamination with emphasis on the
unloading part of the indentation process and the asso-
ciated normal delamination. The interfacial failure during
the loading part has been studied by the authors in a
previous work.8 Delamination was found to occur in
a tangential mode driven by the shear stress at the inter-
face. It is initiated at a radial distance which is two or
three times the contact radius resulting in a ring-shaped
delaminated area and imprinted on the load–
displacement curve as a kind.8 In this paper we will study
the characteristics of normal delamination, conditions
for the occurrence/suppression this mode of failure, and
its fingerprint on the load–displacement curve and pro-
vide some quantitative measures about the interfacial
strength. The effect of residual stress in the film and
waviness of the interface on delamination will also be
investigated. It is emphasized that the calculations as-
sume that other failure events, mainly through-thickness
coating cracks, do not occur. FIG. 1. Geometry of the analyzed problem.

J. Mater. Res., Vol. 16, No. 5, May 2001 1397


The coating is assumed to be a strong, perfectly elastic
material with Young’s modulus Ec and Poisson’s ration
c (subscript c for coating).
The substrate is supposed to be a standard isotropic
elastoplastic material with plastic flow being controlled
by the von Mises stress. For numerical convenience,
however, we adopt a rate-sensitive version of this model,
expressed by

3 sij . p n
. .p . e
ij
p
= = y , (5)
2 e y

. . .e
for the plastic part of the strain rate, p ij ij ij.
Here, sij are the deviator components of the Piola–
.
Kirchhoff stress ij and ij are the dual Lagran-
gean strain-rate components. Furthermore, e
√(3/2)sijsij is the von Mises stress, n is the rate sensitivity
.
exponent, and y is a reference strain rate. In the limit of
n → , this constitutive model reduces to the rate-
independent von Mises plasticity with yield stress y.
Values of n on the order of 100 are frequently used for
metals (see e.g., Ref. 10), so that the value of e at yield
is within a few percent of y for the strain rates that are
encountered in our analysis. The elastic part of the strain
.
rate, e , is given in terms of the Jaumann stress rate as
ij

ij .e
Rijkl kl , (6)

with the elastic modulus tensor Rijkl being determined by
Young’s modulus Es and Poisson’s ration s (subscript s
for substrate).
The problem actually solved is illustrated in Fig. 1.
FIG. 2. Normal and tangential responses according to the interfacial The indenter is assumed rigid and to have a spherical tip
potential [Eq. (1)]: (a) normal response Tn( n ); (b) tangential response characterized by its radius R. The film is characterized by
Tt( t). Both are normalized by their respective peak values max and
its thickness t and is bonded to a half-infinite substrate
max.
by an interface specified above. Assuming both coating
and substrate to be isotropic, the problem is axisymmet-
(tangential) work of separation, n ( t ), can now be ex- ric, with radial coordinate r and axial coordinate z in the
pressed in terms of the corresponding strengths max indentation direction. The actual calculation is carried
( max) as out for a substrate of height L − t and radius L, but L is
taken large enough so that the solution is independent of
1 L and thus approaches the half-infinite substrate solution.
n = exp 1 max n t = exp 1 max t . (3) The analysis is carried out numerically using a finite
2
strain, finite element method. It uses a total Lagrangian
Using these along with the definition q = t / n, we can formulation in which equilibrium is expressed in terms of
relate the normal and shear strengths through the principle of virtual work as

1
dv + dS = (7)
t
max = max . (4) ij
ij T ti ui ds .
v Si v
q 2 exp 1 n

The coupling parameters r and q are chosen such that the Here, v is the total L × L region analyzed and v is its
shear peak traction decreases with positive n and in- boundary, both in the undeformed configuration. With
creases with negative n [Fig. 2(b)]. More details are xi (r, z, ) the coordinates in the undeformed configu-
given in Ref. 8. ration, ui and ti are the components of displacement and

1398 J. Mater. Res., Vol. 16, No. 5, May 2001


traction vector, respectively. The virtual strains ij cor- The substrate is simply supported at the bottom, so
respond to the virtual displacement field ui via the strain that the remaining boundary conditions read
definition,
uz r, L = 0 for 0 r L
1 ur 0, z = 0 for 0 z L . (12)
ij = ui, j + uj,i + uk,iuk, j (8)
2 However the size L will be chosen large enough that
where a comma denotes (covariant) differentiation with the solution is independent from the precise remote
respect to xi. The second term in the left-hand side of conditions.
Eq. (7) is the contribution of the interface, which is here
measured in the deformed configuration (Si {r|z = t}). III. MODEL PARAMETERS
The (true) traction transmitted across the interface has
components T , while the displacement jump is , with There are various material parameters that enter the
being either the local normal direction ( n) or the problem, but the main ones are the interfacial normal
tangential direction ( t) in the (r, z)-plane. Here, and strength max, the coating thickness t, the coating
in the remainder, the axisymmetry of the problem is ex- Young’s modulus Ec, the maximum indentation depth
ploited, so that u t i
0. hmax, and the substrate yield strength y. In the results to
i
The precise boundary conditions are also illustrated in be presented subsequently we focus mainly on the effect
Fig. 1. The indentation process is performed incremen- of the interfacial normal strength max, keeping the same
. value of y 1.0 GPa (with a reference strain rate of
tally with a constant indentation rate h . Outside the con- .
tact area with radius a in the reference configuration, the y 0.1 s−1 and n 100). The elastic properties are
film surface is traction free, taken to be Ec 500 GPa, c 0.33, Es 200 GPa,
and s 0.33.
tr(r, 0) t z(r, 0) 0 for a r L . (9) For the cohesive surface we have chosen the same
Inside the contact area we assume perfect sliding condi- values for n and t, namely 0.1 m. As in the previous
tions. The boundary conditions are specified with respect study,8 the coupling parameters r and q are both taken
to a rotated local frame of reference ( , , ) as shown in equal to 0.5 which give rise to qualitatively realistic cou-
. pling between normal and tangential responses of the
Fig. 1. In the normal direction, the displacement rate u
is controlled by the motion of the indenter, while in the interface. The values of max that have been investigated
tangential direction the traction t is set to zero; i.e., vary approximately between 0.5 and 2.0 GPa. These cor-
respond to interfacial energies for normal failure ranging
u (r, z)
˙ ˙
hcos , t (r, z) 0 for 0 r a . (10) from 150 to 600 J/m2, which are realistic values for the
interface toughnesses of well-adhering deposited films.11
Numerical experiments using perfect sticking conditions
Note that a constant value of q implies that the shear
instead have shown that the precise boundary condi-
strength max always scales with the normal strength
tions only have a significant effect very close to the
max according to Eq. (4).
contact area and do not alter the results for delamination
We have used an indenter of radius R 25 m and
to be presented later. During the loading part, contact
most of the results are for a film thickness t 2.5 m.
nodes are identified by their spatial location with respect
Indentation as well as retraction are performed at a con-
to the indenter; simply, at a certain indentation depth h .
stant rate h ±1 mm/s. The size L of the system ana-
and displacement increment h, the node is considered to
lyzed (Fig. 1) is taken to be 50t. This proved to be large
be in contact if the vertical distance between the node and
enough that the results are independent of L and therefore
the indenter is not greater than h. During the unloading
identical to those for a coated half-infinite medium. The
part, a node is released from contact on the basis of both
mesh is an arrangement of 12,000 quadrilateral elements
its spatial location and the force it exerts on the indenter;
and 12,342 nodes. The elements are built up of four lin-
if the normal component of the nodal force is smaller
ear strain triangles in a cross arrangement to minimize
than a critical value, and the vertical distance between the
numerical problems due to plastic incompressibility. To
node and the indenter is positive, the node is released
resolve properly the high stress gradients under the
from contact. The critical value for the nodal force is
indenter and for an accurate detection of the contact
taken to be 1% of the average current nodal force. It
nodes, the mesh is made very fine locally near the con-
should be noted that using a value 1 order of magnitude
tact area with an element size of t/10.
smaller did not significantly affect the results. The in-
Consistent with the type of elements in the coating
dentation force F is computed from the tractions in the
and the substrate, linear two-noded elements are used
contact region,
along the interface. Integration of the cohesive surface
a contribution in Eq. (7) is carried out using two-point
F= t z r, 0 2 r dr . (11)
0 Gauss integration. Failure, or delamination, of the



interface at any location develops when exceeds . A is a result of the resistance of the substrate to the film
practical definition of when a complete crack has formed bending in this region. It was demonstrated by the au-
is 2 .12 thors8 that the normal displacement induced by this stress
The maximum indentation depth applied in all calcu- will reduce the interfacial shear strength [Fig. 2(b)],
lations is hmax 2t. Further indentation can be done but which in turn may lead to shear delamination.
was not considered relevant since real coatings will have As the indenter is withdrawn, at the same rate as dur-
cracked by then and the present model is no longer ing loading, the elastically bent coating tends to seek its
applicable. original flat shape. For the material parameters here this
peeling tendency induces reverse plastic flow in the sub-
strate under the indenter. As this proceeds, the initially
IV. RESULTS AND DISCUSSION compressive stress evolves into a tensile stress in the
A. Perfect interface interface directly under the initial contact region (Fig. 3).
For the purpose of reference, we first consider a sys- The figure also shows that the tensile area increases
tem with a perfect interface; i.e., its strength is suffi- slowly in size during the process of unloading, and its
ciently higher than the stresses induced by the particular final size is roughly the same as the maximum contact
loading. This can be achieved by rigidly connecting the radius amax.
coating to the substrate, which corresponds to taking To study the evolution of the tensile normal stress at
the interface, its maximum value max is recorded to-
max/ y → . Of particular relevance here, is the devel- n

opment of the stress distribution along the interface dur- gether with its position r along the interface, as shown in
ing the unloading stage and, in particular, the component Fig. 4. In the initial stages of unloading, tension is found
normal to the interface n. From this, we can already get only in the ring outside the contact area (Fig. 3). Upon
qualitative insight into when and where delamination continued unloading, the peeling effect causes interfacial
may occur. tension to develop rapidly, Fig. 4(a), with the location of
Figure 3 shows the normal stress at the interface at the maximum closely following the instantaneous con-
different instants between maximum indentation depth tact radius a [Fig. 4(b)]. The largest value of max n

and complete retraction of the indenter, as specified 2.7 GPa obtained in this particular case is reached at the
through the load F relative to the maximum indenter end of the unloading and located at the symmetry axis.
load. At the maximum indentation depth, the interface On the basis of these results, interfacial failure leading
stress is of course compressive and almost uniform over to normal delamination may be expected during the un-
the current contact area due to plastic flow in the sub- loading stage when the interfacial strength max is lower
strate. The compressive stress attains a peak value of than the maximum tensile stress max reached at any
n

approximately 4 GPa just outside the contact region moment. In the present case, normal delamination is
of radius amax. Relatively low tensile normal stresses are avoided on the other hand if the interfacial strength max
found beyond the compressive region, at r ≈ 3amax. This exceeds 2.7 GPa.
Figure 5, curve (e), shows the indentation load versus
displacement curve for this case of a perfect interface.
Such a curve is one of the most common outputs of
indentation experiments. Its importance stems from the
fact that it is a signature of the indented material system.
Several techniques have been reported in the literature to
extract the mechanical properties of both homogeneous
and composite or coated materials from indentation ex-
periments (e.g., Refs. 13–17). In the forthcoming section,
we will therefore study the interfacial failure process in
more detail and provide some qualitative measures of the
interfacial strength.

B. Finite-strength interface
In this section, and throughout the rest of this paper,
we will study interfaces with finite strengths to allow for
interfacial delamination to develop. To demonstrate the
FIG. 3. Normal stress variations along a perfect interface at the be- effect of the interfacial failure on the load–displacement
ginning of unloading (F = Fmax) until complete retrieval of the in- data, Fig. 5 shows the predicted curves for different val-
denter (F 0). ues of interfacial strength max. The rest of the material



FIG. 5. Load versus displacement curves for several values of inter-
facial strength max: (a) 0.55; ( b) 1.1; (c) 1.5; (d) 2.2 GPa. Curve (e)
is for a perfect interface.

reduction is due to shear delamination at that stage, as
discussed in detail in Ref. 8. In all other cases shown, the
interface strength was large enough to prevent shear de-
lamination but not normal delamination.
The interfacial strength above which delamination is
prevented is found to be max 2.21 GPa (curve d in
Fig. 5). From the results discussed above for a perfect
interface, however, we expected delamination at even
higher strengths, up to 2.7 GPa. The difference must be
attributed to the fact that the cohesive surface description
for the finite-strength interface provides additional com-
pliance to the system even before failure. This additional
FIG. 4. (a) Evolution of the maximum normal stress max with inden-
compliance results from the limited normal opening at
n
tation depth during unloading. (b) Corresponding location at the in- the interface ( n < n), whereas a perfect interface, by
terface at which the stress is maximum. definition, does not allow such opening. Although the
energy consumed at the interface in this state is ex-
tremely small, the extra compliance does give rise to a
and geometrical parameters are the same as before. In- small redistribution of the normal stress over the inter-
terfacial delamination during unloading was found in all face and a reduction of the maximum normal stress max. n
cases shown in Fig. 5 (except case e). Compared to the Figure 6(a) shows a contour plot of the von Mises
perfect interface case (curve e), the initiation of delami- effective stress at the end of the loading stage (F = Fmax)
nation is seen to result in a rather sudden reduction of the for the case (c) in Fig. 5 with max 1.5 GPa. The size
unloading stiffness at sufficiently small F. For higher of the plastic zone at this depth of h 2t is about 5 times
interfacial strengths, delamination is imprinted on the the maximum contact radius. To illustrate the delamina-
load versus displacement curve as a hump where tion process, Fig. 6(b) shows a contour plot of the verti-
the stiffness becomes negative. This phenomena will be cal stress component zz at the end of the unloading
explained in more detail later in this section. Another process (F 0). The first thing to observe is that the
characteristic of delamination that can be observed in the radius of the delaminated zone, rd, is about 50% larger
load–displacement curve is the negligible residual inden- than the maximum contact radius amax reached during
tation depth at the end of the unloading. In the absence of indentation. Second, we observe a region with compres-
delamination (case e), the residual indentation depth is sive normal stress in front of the delamination tip. This
more than half the maximum indentation depth. Curve a, region is the remainder of the compressive region gen-
which corresponds to the lowest max, shows a little de- erated during the loading stage, which has apparently
crease in the stiffness at the end of the loading stage. This hardly changed during unloading. It thus seemed that



FIG. 7. Evolution of the delamination radius during unloading for
hmax 5 m and several values of max (or equivalently n).

at a relatively high initial propagation velocity compared
.
to the indentation rate h and then reaches a lower ve-
.
locity on the order of h . The crack is stopped when it
reaches the region with sufficiently high compressive
stress (Fig. 6). The final delamination radius is about
1.5 times the maximum contact area for all values of
max. It is clear in the figure that for lower interfacial
strengths, delamination starts earlier in the unloading
process. On the other hand, the lower the interfacial
strength, the lower the residual indentation depth hr (per-
manent indentation depth left at the end of the unload-
ing). Figure 7 reveals that residual indentation depth hr
for several values of max 0.55 GPa. Lower interfacial
strengths even lead to small negative residual indentation
depths, where the coating bulges upwards at the end of
the unloading.
The observations indicate that delamination is the out-
come of a complex interaction between various mecha-
FIG. 6. (a) Contour plot of the von Mises stress at the end of loading
nisms. To get further insight into this competition,
(F = Fmax). (b) Contour plot of the stress component zz at the end of
the unloading (F 0) for max 1.5 GPa (curve c in Fig. 5). The plot Fig. 8(a) shows the decomposition of the total energy of
also shows the delaminated region. the system into interfacial energy Uin, elastic energy Uel
(in the film and substrate), and dissipated, plastic energy
delamination was initiated under the retrieving indenter, Upl for the case of max 1.5 GPa (curve c in Fig. 5).
expanded in the radial direction and was arrested in this Other values of interfacial strength show the same quali-
compressive interfacial stress region. tative behavior. In this particular case, delamination ini-
The progressive development of delamination with tiated at h 1.5t 3.75 m. It is clear in the figure that
continued unloading is shown in Fig. 7 for several values the plastic energy is constant at the initial stage of the
of max. It should be noted that, except for max unloading, i.e., the initial stage for the unloading is al-
2.2 GPa, delamination starts at a distance from the sym- most purely elastic. This is in agreement with what is
metry axis. For these cases rd represents the location of commonly observed in indentation experiments Ref. 13.
the delamination tip which is traveling away from the Limited reverse plasticity is seen to have contributed to a
symmetry axis. Since the other tip reaches the symmetry little increase (less than 10%) in the plastic energy. At the
axis almost immediately, rd can be considered to a good onset of delamination, the plastic energy reaches a con-
approximation as the radius of the delaminated circular stant value. The contribution of the film and the substrate
area. In all cases shown in Fig. 7 delamination starts to the elastic energy is demonstrated in Fig. 8(b). The



stiffness, shown in Fig. 7, is now readily attributed to the
spontaneous opening of the interface at the initial stage of
delamination (Fig. 7). As explained in the previous para-
graph and shown in Fig. 8, the processes that control the
system during delamination are the unflexing of the coat-
ing and the interfacial delamination. The coating evi-
dently provides a positive contribution to the overall
stiffness, whereas the energy release from the interface
gives a negative contribution. This can be seen in Fig. 8,
where the stiffness provided by each energy source is the
curvature of the corresponding curve. For relatively
strong interfaces, the energy release from the interface
dominates during the first stage of delamination when the
.
rate of propagation, relative to the indentation rate h , is
high. During the second stage, the process is governed by
the unflexing of the coating, thus giving rise to a positive
overall stiffness (note that the coating response is con-
strained by the indenter which is withdrawn at a given
rate). It is this complex interplay between these two
terms which shapes the overall behavior of the system,
including the load–displacement curve.

C. Comparison with a simple estimate
Deduction of quantitative information about the inter-
facial strength from indentation experiments, in particu-
lar from load–displacement curves and delamination
areas, is hindered by the rather complicated interplay
between the film elastic energy and the interfacial en-
ergy. A simple estimate for the work of interfacial
debonding from final delamination results has been given
by Hainsworth et al.6 This estimate is based on an energy
balance involving the interfacial energy and the elastic
FIG. 8. (a) Decomposition of total energy into interfacial energy (Uin),
energy in the coating (the elastic energy in the substrate
elastic energy (Uel), and plastic energy (Upl ). (b) Contribution of the
film and substrate to the elastic energy. In (a) and (b) max 1.5 GPa, is neglected). The latter is approximated by the elastic
the normalization constant is Umax ∫hmaxF dh, and the vertical dashed
0
energy of a centrally loaded disc of radius rd with
lines identify the initiation of delamination. clamped edges. On the basis of this model, the interfacial
work of separation is estimated by
elastic energy of the substrate is seen to decrease more
rapidly compared to the elastic energy of the film at the 2Ect 3 hmax2 − hr2
initial stage of the unloading. This is in agreement with
est
n = 2
, (13)
31 c rd4
what is reported in the literature that the initial stiffness
of the unloading is predominantly controlled by the sub- in terms of directly measurable quantities.
strate for indentation depths larger than the film thick- As the model shows a strong dependence on the coat-
ness.13–17 At the onset of delamination, the substrate ing thickness t and the maximum indentation depth hmax,
elastic energy reaches a constant value, whereas the we have chosen to vary these two parameters over a
film elastic energy decreases as the film unbends. This certain range and compare the model predictions with
indicates that the main contribution to the energy release, our FEM findings. A set of calculations using a conical
and hence the advance of delamination, come from the indenter with a 68° semiangle is also performed to ex-
film. It is also interesting to notice that, at the end of amine the model’s sensitivity to the indenter’s geometry
the unloading, there still exists some elastic energy in the which is not captured by Eq. (13).
system. This energy is small compared to the dissipated Despite its very approximate nature, Eq. (13) does
energy (plastic energy), but when compared with the in- capture some of the qualitative trends, as shown in Fig. 9.
terfacial energy, Uin, it seems to have a significant value. For instance, one expects from (13) that rd2 hmax for a
On the basis of the above observations, the unstable given interfacial strength (or energy) and coating prop-
part of the load–displacement curves, with negative erties (and neglecting the residual indentation depth).



The results of a series computations for two different The more serious limitation of Eq. (13) is that the in-
strengths are summarized in Fig. 9(a) and are seen to be terfacial energy estimated from the numerical results do
consistent with this scaling. The conical indenter results not agree quantitatively with the actual energies. As dem-
presented in the figure show the same trend. Sanchez ´ onstrated in Tables I–III, the interfacial energies are se-
et al.7 have used Eq. (13) and a modified version of it on verely overestimated. In Table I we notice that the higher
their cross-sectional indentation data, and they have also the maximum indentation depth, the better the estimate.
confirmed the linear relation between the delamination This can be understood by recalling that the model is
area and the maximum deflection of the coating. Second, based on the expression for the deflection of a clamped
according to (13), rd4/3 is proportional to t, with all other disc loaded at the center,18 where the deformation is
quantities being the same. Our results, shown in Fig. 9(b) assumed to be pure bending. The contribution of the
are consistent with this as well. Finally, over the range of stretching is ignored; this is reasonable when the radius
Ec 350–600 GPa, the proportionality between rd4 of the disc is large compared to its thickness. In the
and Ec is also found to be consistent with the prediction case of indentation, this condition is analogous to contact
of Eq. (13). radius (or maximum indentation depth) being larger than
However, not all trends are correct. For example, the film thickness. This explains the better estimation at
Eq. (13) predicts a lower slope for the delamination area larger maximum indentation depths. This trend is also
versus h max curve for higher values of interfacial observed for the conical indenter in Table II, but the
strength, whereas the FEM results presented in Fig.9(a) quality of the estimate here is even worse. The reason is
show the opposite tendency. that the cone produces more stretching of the film than
the sphere, resulting in less accuracy of the model. In
Table III, the smaller the coating thickness, the better the
estimate according to Eq. (13). The same explanation

TABLE I. Estimates for n from Eq. (13) on the basis of the com-
puted values hr and rd for t 2.5 m and several values of hmax. The
actual value is n 500 J/m2.
est
hmax ( m) hr ( m) rd ( m) n / n

2.5 0.79 8.14 15.02
3.0 0.67 10.62 7.86
3.5 0.64 12.25 6.15
4.0 0.64 13.61 5.32
4.5 0.63 14.82 4.82
5.0 0.63 15.94 4.46

TABLE II. Same as in Table I but for a conical indenter.
est
hmax ( m) hr ( m) rd ( m) n / n

2.5 0.43 4.75 139.34
3.0 0.42 6.49 58.24
3.5 0.43 7.81 37.92
4.0 0.45 9.03 27.84
4.5 0.47 10.24 21.32
5.0 0.49 11.45 16.86

TABLE III. Estimates for n from E8. (13) on the basis of the com-
puted values of hr and rd for hmax 5 m and several values of t. The
actual value is n 500 J/m2.
est
t ( m) hr ( m) rd ( m) n / n

2.5 0.63 15.94 4.46
3.0 0.58 16.59 6.58
3.5 0.56 17.23 9.00
4.0 0.54 17.84 11.69
4.5 0.54 18.43 14.62
FIG. 9. (a) Delamination area rd2 versus the maximum indentation 5.0 0.54 19.00 17.75
depth hmax. (b) rd4/3 versus coating thickness t.



holds here too. Evidently, the assumption that the disc is
clamped at its boundary in questionable. If it is assumed
that the disc is simply supported, the expression for est n
in Eq. 13 must be multiplied by (1 + c)/(3 + c).
This will give better estimates, but large errors are still
possible.
Note that Eq. 13 does not incorporate the influence of
the substrate. To see the accuracy of this approximation,
we have investigated the dependence of the delamination
radius rd on the substrate properties Es and y. Varying
the substrate Young’s modulus Es from 100 to 500 GPa,
the resulting delamination radius increases with Es by
25%. On the other hand, an increase of the yield stress y
from 0.72 to 2.0 GPa gives values of rd that decrease by
only 6%. The reason for this is that the yield stress de-
termines the size of the plastic zone in the substrate but
not the permanent deformation immediately below the FIG. 10. Load– displacement curves for several values of hmax, for a
indenter; the latter is what is controlling the delamination coating strength of max 1.85 GPa.
radius. However, it should be noted, as will be discussed
in the next section, that the yield stress plays a major
For values of hmax less than the coating thickness (t
role in determining whether delamination will take
2.5 m), c max shows a relatively rapid increase,
place at all.
Fig. 11(a). This increase is attributed to the increase in
D. Critical value of interfacial strength the bending moment in the coating. The bending moment
for delamination is proportional to the curvature of the coating which in-
creases rapidly with the indentation depth until the coat-
Whether or not delamination takes place depends on
ing takes the shape of the indenter. After that point, the
the tensile normal stress that can be generated at the
curvature does not change much but the bent region
interface during the unloading process. The ultimate
propagates outward, and this corresponds to the slower
value of this stress relative to the interface strength max
increase in c max for higher indentation depths.
depends on almost all parameters involved in the bound-
Figure 11(b) shows also an initial rapid increase in
ary value problem in a rather complex way. We have
the critical strength with the coating thickness due to the
performed a parameter study involving the coating elas-
increase of the bending moment with t3. For thicker coat-
tic modulus, the substrate yield stress, the maximum in-
ings, the critical strength decreases due to the decrease in
dentation depth, and the coating thickness. For each
the coating curvature because the substrate becomes rela-
parameter combination, delamination is suppressed if the
tively softer. Figure 11(c) shows an almost linear in-
interfacial strength is higher than a critical value of c .
max
crease of the critical strength with the coating Young’s
As an example, Fig. 10 shows load–displacement
modulus. The increase of the critical strength with the
curves for different values of maximum indentation
substrate yield stress y is shown in Fig. 11(d). This in-
depths. Delamination is seen to occur if hmax is above a
crease is caused by the reverse plasticity that takes place
certain critical value, and it is recognized by the hump
prior to delamination (Fig. 8). The higher the yield stress,
left on the curve and the negligible residual indentation
the higher the stresses which can be reached at the sub-
depth. Lower indentation depths do not create normal
strate. Since the normal stress is continuous across the
stresses that exceed the interfacial strength max and,
interface, higher tensile normal stress can be reached
therefore, do not lead to delamination.
with increasing y, thus making it possible to delaminate
Figure 11 shows the variation of the critical strength
c stronger interfaces.
max with (a) the maximum indentation depth, (b) the
coating thickness, (c) the coating Young’s modulus, and
(d) the substrate yield stress. Higher values of the coating E. Residual stresses and interfacial waviness
Young’s modulus Ec, the coating thickness t 3, or the Coated systems generally contain residual stresses.
maximum indentation depth hmax lead to delamination of These are due to the deposition process itself, to the
stronger interfaces. These are explained by the fact that thermal expansion mismatch between the coating and
the driving force for delamination is the unbending of the the substrate, or a combination of the two. To study the
coating. Despite the limitations of the circular disc model influence of residual stresses on delamination, we
pointed out before, these trends are roughly consistent have introduced uniform in-plane stress in the film prior
with Eq. (13) but not when looked at in more detail. to indentation. This has been achieved, for numerical



c
FIG. 11. Critical value of the interfacial strength max versus (a) hmax, (b) t, (c) Ec, and (d) y.

convenience, by assigning different thermal expansion valleys and crests where the normal stress component has
coefficients to coating and substrate and by subjecting a local maximum. Neighboring delaminated areas link up
the system to various temperature changes to generate before the delamination front propagates to the next
stresses ranging from −10 GPa (compressive) to 10 GPa crest/valley. Even though the precise evolution of de-
(tensile). Subsequently, we perform the indentation cal- lamination depends on the waviness of the interface, for
culations as before.
Compressive stress in the coating is found to delay the
delamination process, or to even prevent delamination,
whereas the opposite happens with tensile stresses. This
is explained by the fact that residual stress will have an
out-of-plane component after the deformation of the
coating. In the case of tensile stress, this component will
tend to enhance the unbending of coating during the un-
loading and, thus, will assist delamination. As a conse-
quence, the critical strength to prevent delamination will
increase with residual tension in the coating. Compres-
sive stress has the opposite effect. For example, a coating
of the default thickness of t 2.5 m with a interfacial
strength of max 1.84 GPa was found earlier to de-
laminate after indentation to h max 5 m [see
Fig. 11(a)], but delamination is prevented under a re-
sidual stress of −10 GPa. The delamination radius rd is
relatively insensitive to the residual stress: over a range
of −7.5 to 10 GPa, rd varies between 14.4 and 16.7 m
compared to rd 15.94 m for the stress-free coating
(cf. Table I).
Roughness of the interface is commonly simplified by
a sinusoidal wave (e.g., Ref. 19). To study the effect of
FIG. 12. Example of normal delamination for a case with a rough
roughness on delamination, a wave of an amplitude up to interface, modeled as a sinusoidal wave with an amplitude of 0.12t and
0.2t and a wavelength up to 2t were introduced along the a wavelength equal to t. In this case, hmax 5 m and max
interface; see Fig. 12. Delamination is found to start at 1.85 GPa.



all cases considered here we did not find a significant The disc model estimate6 has been compared with our
effect on the critical indentation depth at which delami- numerical findings for a range of parameters. It does
nation starts nor on the final delamination radius. capture some of the qualitative aspects of delamination.
But, it tends to strongly overestimate the interfacial
strength or energy of separation.
V. CONCLUSIONS Critical values of the interfacial strength were calcu-
For the purpose of studying interfacial delamination, lated for several parameter combinations. The general
numerical simulations have been carried out of the in- trends of the variation of these critical values with the
dentation process of a coated material by a spherical involved parameters are easily interpreted, whereas
indenter. To describe interfacial failure, the interface be- the details of this variation are governed by the nonlinear
tween the film and the substrate was modeled by means nature of the problem.
of a cohesive surface, with a coupled constitutive law for Compressive residual stress in the film delays delami-
the normal and the tangential response. Failure of the nation, and if high enough, it might even prevent
interface by normal or tangential separation, or a combi- delamination, whereas tensile residual stress has an op-
nation, is embedded in the constitutive model and does posite effect. Waviness of the interface was not found to
not require any additional criteria. have a significant effect on delamination. Both conclu-
Normal delamination occurs during the unloading sions, however, are intimately tied to the assumption that
stage of the indentation process. A circular part of the the coating remains intact during indentation.
coating, directly under the contact area, is lifted off from
the substrate, driven by the bending moment in the coat-
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